[br][table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/z8r7fkjb][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAA2CAYAAABA3FA2AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACpSURBVGhD7dkxCsJAFEXR/wZiJWJhIW7MUnApriwLEFdhZy0iiN8M2tjdLr94h8wEUt3yQaRhyMiMiH7mpulRv0vU/Gm/dymOohxFOYpyFOUoylGUoygdd9tye0qv1upFTUVenoSjKEdRjqIcRTmKchTlKErX/abgyLvUG3nKs5cn4ijKUZSjKEdRjqIcRRWN6j9six2dxkMu9YiV7t+Ps8l45iJu73V8AE/fHKUjFbbZAAAAAElFTkSuQmCC[/img][/url][/td][td][size=50] this activity is a page of [color=#980000][i][b]geogebra-book[/b][/i][/color][br] [url=https://www.geogebra.org/m/nzfg796n#material/hhnfvjkv][color=#0000ff][u][i][b]elliptic functions & bicircular quartics & . . .[/b][/i][/u][/color][/url]([color=#ff7700][i][b]28.04.2023[/b][/i][/color])[/size][/td][/tr][/table][size=50][i][b][size=50][right][size=50]this activity is also a page of [/size] [color=#980000]geogebra-book[/color] [url=https://www.geogebra.org/m/xtueknna][color=#0000ff][u]geometry of some complex functions[/u][/color][/url] [color=#ff7700]october 2021[/color][/right][/size][/b][/i][/size]

[size=85]A [size=100][b][i][color=#ff0000]parabolic pencil[/color][/i][/b][/size] of [b][i][color=#ff0000]circles[/color][/i][/b] consists of all [b][color=#ff0000]circles[/color][/b] that touch a given [b][i][color=#ff0000]circle[/color][/i][/b] at a given [b][i][color=#0000ff]point [/color][/i][/b][br]- the [b][i][color=#0000ff]base point[/color][/i][/b] of the [b][i][color=#ff0000]pencil[/color][/i][/b]. [br][br]The [b][i][color=#ff00ff]parallels[/color][/i][/b] to the[/size] [math]x[/math][size=85]-axis are such a [b][i][color=#ff0000]parabolic[/color][/i][/b] "[b][i][color=#ff0000]pencil[/color][/i][/b] of [b][i][color=#ff0000]circles[/color][/i][/b]" in terms of [b][i][color=#0000ff]möbius geometry[/color][/i][/b]: [br]The "[b][i][color=#ff0000]circles[/color][/i][/b]" here are [b][i][color=#0000ff]straight lines[/color][/i][/b], which pass through the [b][i][color=#0000ff]point[/color][/i][/b][/size] [math]\infty[/math] [size=85]and touch there. [br]This is best recognised with the help of the [b][i][color=#38761d]stereographic projection[/color][/i][/b].[br]Each [b][i][color=#ff0000]pencil[/color][/i][/b] of [b][i][color=#0000ff]parallels[/color][/i][/b] is a [b][i][color=#ff0000]parabolic pencil [/color][/i][/b]of [b][i][color=#ff0000]circles[/color][/i][/b] with as its base point [math]\infty[/math].[br][br]A [b][i][color=#0000ff]Möbius transformation[/color][/i][/b], which maps the [b][i][color=#0000ff]points[/color][/i][/b][/size] [math]0,1,\infty[/math] [size=85]to three different points[/size] [math]w_1,w_2,w_3[/math], [br][size=85]transforms the [b][i][color=#ff0000]pencil[/color][/i][/b] of [b][i][color=#0000ff]straight lines[/color][/i][/b] parallel to the[/size] [math]x[/math][size=85]-axis into a [b][i][color=#ff0000]parabolic pencil[/color][/i][/b] of [b][i][color=#ff0000]circles[/color][/i][/b], which [br]maps the [b][i][color=#0000ff]parallels[/color][/i][/b] onto [b][i][color=#ff0000]circles,[/color][/i][/b] which touch the [i][b][color=#ff0000]circle[/color][/b][/i] through[/size] [math]0,1,\infty[/math] [size=85]in[/size] [math]\infty[/math].[br][size=85]Conversely, every [b][i][color=#ff0000]parabolic pencil[/color][/i][/b] of [b][i][color=#ff0000]circles[/color][/i][/b] can be transformed into a [b][i][color=#ff0000]pencil[/color][/i][/b] of [b][i][color=#0000ff]parallel lines[/color][/i][/b] [br]by a [b][i][color=#0000ff]Möbius transformation[/color][/i][/b]. [br][/size]

[size=85]In general, [b][i][color=#ff0000]pencils of circles[/color][/i][/b] and their [b][i][color=#9900ff]loxodromes [/color][/i][/b]- i.e. the curves, [br] which intersect the [b][i][color=#ff0000]circles[/color][/i][/b] of the [b][i][color=#ff0000]pencil[/color][/i][/b] at a constant [b][i][color=#38761d]angle[/color][/i][/b] - [br]are characterised by a [b][i][color=#ff00ff]differential equation[/color][/i][/b] and thus by a [b][i][color=#ff00ff]vector field[/color][/i][/b] of the type[br][list][*][math]g'=c\cdot\left(g-f_1\right)\cdot\left(g-f_2\right)\mbox{ mit }f_1,f_2,c\in\mathbb{C}[/math].[br][/*][/list]Here the [b][i]complex solution function[/i][/b] is analytical, or meromorphic. [br]The [b][i]zeros[/i][/b], which we call [b][i][color=#00ff00]focal points[/color][/i][/b], can coincide ( - then there is a [b][i][color=#ff0000]parabolic pencil of circles[/color][/i][/b] - ).[br]One can interpret the [b][i][color=#ff0000]circles[/color][/i][/b] of a [b][i][color=#ff0000]hyperbolic[/color][/i][/b] [b][i][color=#ff0000]pencil[/color][/i][/b] dynamically as [b][i][color=#0000ff]circular waves[/color][/i][/b], which propagate [br]from a source in the direction of the [b][i][color=#ff0000]circles[/color][/i][/b] of the [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]elliptic pencil[/color][/i][/b]. [br]The source and the sink are the [b][i][color=#00ff00]focal points[/color][/i][/b] of the [b][i][color=#0000ff]wave motion[/color][/i][/b].[br]We call these [b][i][color=#38761d]vector fields[/color][/i][/b] linear. [br]For explanation we refer to the representation of the [b][i][color=#0000ff]Möbius group[/color][/i][/b] by the complex [b]special orthogonal group[/b] [b]SO(3,[/b][math]\mathbb{C}[/math][b])[/b][br]and its [b]LIE algebra[/b] [math]\mathbf{\mathcal{so}\left(3,\mathbb{C}\right)[/math]. [math]\hookrightarrow[/math] [color=#980000][i][b]geogebra-book[/b][/i][/color] [color=#0000ff][i][b]Möbiusebene[/b][/i][/color], [/size][size=85]especially the[/size][size=85] chapt. [url=https://www.geogebra.org/m/kCxvMbHb#chapter/168949][color=#0000ff][u][i][b]Kreisbüschel und lineare Vektorfelder[/b][/i][/u][/color][/url].[br][br]If [b][color=#cc0000]2 [/color][/b]such [b][i][color=#134f5c]linear vector fields[/color][/i][/b] are superimposed, "quadratic vector fields" are obtained, whose solution curves [br]are [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]conic sections[/color][/i][/b] or [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]bicircular quartics[/color][/i][/b]. [br][b][i][color=#00ff00]Focal points[/color][/i][/b] are in each case the zeros of the [/size][size=85][b][i][color=#134f5c]linear vector fields[/color][/i][/b][/size][size=85].[br]In these cases, the solution curves are [b][i][color=#38761d]angle[/color][/i][/b] bisectors of the intersecting [b][i][color=#ff0000]circles[/color][/i][/b] from the two [b][i][color=#ff0000]pencils of circles[/color][/i][/b].[br][br]links: [br][math]\hookrightarrow[/math] [color=#980000][i][b]geogebra-book[/b][/i][/color] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/kCxvMbHb]möbiusebene[/url][/b][/i][/u][/color][br][math]\hookrightarrow[/math] [/size][size=85][size=85][color=#980000][i][b]geogebra-book[/b][/i][/color][/size] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/fzq79drp]Leitlinien und Brennpunkte[/url][/b][/i][/u][/color][br][br][/size]